The heightened sensitivity can be explained as commonly occuring during times of imminent harm. For example, sometimes when you fall down, it seems as if you are moving in slow motion, aware of everything happening around you, preparing yourself for when you hit the ground! In the same way Farquar seems suspended from time, almost standing outside of the incident, objective enough to notice details. Though this sequence is all in his imagination, his sensitivity to his surroundings was intense leading up to the hanging as well. An example would be his focusing on the floating driftwood, the seemingly loud ticking of his watch in para. 5. He knew he was about to lose all sensations, so each one was on keen alert to take advantage of his last moments.
All of Farquhar's sensitivity is really related to his imagination and the reality that he is suffocating at the end of a hangman's noose. When he feels the rope first break, he also feels a sharp pain in his neck and a feeling of suffocation. After he surfaces from the water, his sensitivity is not only elaborate but also impossible. He can see the color of the eyes of the sharpshooter. He escapes and entire volley of gunshots plus cannon fire from the Union army. These events can be realistically explained only as the imaginations of a dying man. The proof comes in the last line of the story: "Peyton Farquhar was dead." He had only imagined the rope breaking, his escape and reuniting with his wife.